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Theorem ofsubeq0 9743
Description: Function analog of subeq0 9073. (Contributed by Mario Carneiro, 24-Jul-2014.)
Assertion
Ref Expression
ofsubeq0  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  ( ( F  o F  -  G
)  =  ( A  X.  { 0 } )  <->  F  =  G
) )

Proof of Theorem ofsubeq0
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simp2 956 . . . . . . 7  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  F : A --> CC )
2 ffn 5389 . . . . . . 7  |-  ( F : A --> CC  ->  F  Fn  A )
31, 2syl 15 . . . . . 6  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  F  Fn  A
)
4 simp3 957 . . . . . . 7  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  G : A --> CC )
5 ffn 5389 . . . . . . 7  |-  ( G : A --> CC  ->  G  Fn  A )
64, 5syl 15 . . . . . 6  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  G  Fn  A
)
7 simp1 955 . . . . . 6  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  A  e.  V
)
8 inidm 3378 . . . . . 6  |-  ( A  i^i  A )  =  A
9 eqidd 2284 . . . . . 6  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( F `  x )  =  ( F `  x ) )
10 eqidd 2284 . . . . . 6  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( G `  x )  =  ( G `  x ) )
113, 6, 7, 7, 8, 9, 10ofval 6087 . . . . 5  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( ( F  o F  -  G
) `  x )  =  ( ( F `
 x )  -  ( G `  x ) ) )
12 c0ex 8832 . . . . . . 7  |-  0  e.  _V
1312fvconst2 5729 . . . . . 6  |-  ( x  e.  A  ->  (
( A  X.  {
0 } ) `  x )  =  0 )
1413adantl 452 . . . . 5  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( ( A  X.  { 0 } ) `  x )  =  0 )
1511, 14eqeq12d 2297 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( (
( F  o F  -  G ) `  x )  =  ( ( A  X.  {
0 } ) `  x )  <->  ( ( F `  x )  -  ( G `  x ) )  =  0 ) )
16 ffvelrn 5663 . . . . . 6  |-  ( ( F : A --> CC  /\  x  e.  A )  ->  ( F `  x
)  e.  CC )
171, 16sylan 457 . . . . 5  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( F `  x )  e.  CC )
18 ffvelrn 5663 . . . . . 6  |-  ( ( G : A --> CC  /\  x  e.  A )  ->  ( G `  x
)  e.  CC )
194, 18sylan 457 . . . . 5  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( G `  x )  e.  CC )
20 subeq0 9073 . . . . 5  |-  ( ( ( F `  x
)  e.  CC  /\  ( G `  x )  e.  CC )  -> 
( ( ( F `
 x )  -  ( G `  x ) )  =  0  <->  ( F `  x )  =  ( G `  x ) ) )
2117, 19, 20syl2anc 642 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( (
( F `  x
)  -  ( G `
 x ) )  =  0  <->  ( F `  x )  =  ( G `  x ) ) )
2215, 21bitrd 244 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( (
( F  o F  -  G ) `  x )  =  ( ( A  X.  {
0 } ) `  x )  <->  ( F `  x )  =  ( G `  x ) ) )
2322ralbidva 2559 . 2  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  ( A. x  e.  A  ( ( F  o F  -  G
) `  x )  =  ( ( A  X.  { 0 } ) `  x )  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
243, 6, 7, 7, 8offn 6089 . . 3  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  ( F  o F  -  G )  Fn  A )
2512fconst 5427 . . . 4  |-  ( A  X.  { 0 } ) : A --> { 0 }
26 ffn 5389 . . . 4  |-  ( ( A  X.  { 0 } ) : A --> { 0 }  ->  ( A  X.  { 0 } )  Fn  A
)
2725, 26ax-mp 8 . . 3  |-  ( A  X.  { 0 } )  Fn  A
28 eqfnfv 5622 . . 3  |-  ( ( ( F  o F  -  G )  Fn  A  /\  ( A  X.  { 0 } )  Fn  A )  ->  ( ( F  o F  -  G
)  =  ( A  X.  { 0 } )  <->  A. x  e.  A  ( ( F  o F  -  G ) `  x )  =  ( ( A  X.  {
0 } ) `  x ) ) )
2924, 27, 28sylancl 643 . 2  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  ( ( F  o F  -  G
)  =  ( A  X.  { 0 } )  <->  A. x  e.  A  ( ( F  o F  -  G ) `  x )  =  ( ( A  X.  {
0 } ) `  x ) ) )
30 eqfnfv 5622 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
313, 6, 30syl2anc 642 . 2  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  ( F  =  G  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
3223, 29, 313bitr4d 276 1  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  ( ( F  o F  -  G
)  =  ( A  X.  { 0 } )  <->  F  =  G
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   {csn 3640    X. cxp 4687    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Fcof 6076   CCcc 8735   0cc0 8737    - cmin 9037
This theorem is referenced by:  psrridm  16149  dv11cn  19348  coeeulem  19606  plydiveu  19678  facth  19686  quotcan  19689  plyexmo  19693  mpaaeu  27355
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-ltxr 8872  df-sub 9039
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